On Reticular Properties of a Baric Algebra

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598 R. BRU AND J. VITbRIA

ON RETICULAR PROPERTIES OF A BARIC ALGEBRA

by SANTOS GONZALEZ JIMENEZ”*‘2

1. Zntroduction The relationship between certain linear nonassociative algebras and the theory of population-genetic models was first investigated by Etherington [3]. He defined the concept of baric, train, and special train algebras, and he showed that many algebras modeling concrete biological situations are special train algebras. Schafer [12] found that not all algebras from genetics are of this type. The popular algebra for simple Mendelian segregation already fails to have this property. He defined the mathematical term “genetic algebra.” Genetic algebras are a subclass of train algebras. In this context we want to note the following statements in J. Roy. Statist. Sot. Ser. B 43:1-19 (1981):

The subject of genetic algebras was, in a sense, my baby. Over the years it was amazing to me to see how my baby has (varying the metaphor) taken root and proliferated. After R. D. Schafer’s seminal paper of 1949, all the best new ideas on the subject have been introduced by Professor Holgate himself. (Professor I. M. H. Etherington, Edinburgh University)

The study of the algebraic structure deriving from the fact that the multiplication rules correspond to the Mendelian segregation mechanism and the range of types of inheritance and breeding structures to which genetic algebras have been applied have both been developed extensively. The use of an algebra in the description of a natural phenomenon is appropriate when a useful interpretation can be given to the processes of adding and multiplying elements. (Professor P. Holgate, London College)

Genetic algebras express the production of a new generation as a simple multiplication: mother X father = child. This has obvious simplicity and elegance. (Professor C. A. B. Smith, University College, London)

Many papers by Gonshor, Holgate, Heuch, W&z-Busekros, Piacentini, Micah, Perez Vargas, and others have appeared about this topic, and nonassociative algebras in genetics have developed into a field of independent mathematical interest. These algebras are in general commutative but nonassociative; furthermore, they do not belong to any of the well-known classes of

“Departmento de Matemiticas, Facukad de Ciencias, Universidad de ikragoza, 5OG@- Zaragoza, Spain. ‘2Partially supported by I.E.R. CONFERENCEREPORT 599 nonassociative algebras, such as Lie, Jordan, or alternative algebras. It is of interest to study the connection with some of the above classes of algebras. In particular Schafer [12] and Holgate [5] proved that the gametic and zygotic algebras for a single locus are Jordan algebras. Also, G. M. Piacentini [13] determined a necessary and sufficient condition under which the gametic algebra of mutation is still a Jordan algebra. Finally Micah and Ouattara [14] and W&z-Busekros [ll] obtained a necessary and sufficient condition for a Bernstein algebra to be a Jordan algebra. The Bernstein algebras are baric algebras in which (X~)~ = WAXY for every element X; they describe the inheritance of populations which are in equilibrium after one generation. These algebras were introduced by Holgate [6], who gave an algebraic formulation of the Bernstein problem. This old problem was posed by Bernstein in 1923 in order to classify all evolutionary operators satisfying the stationarity principle, i.e., all idempotent quadratic operators acting on the stochastic simplex. The principle of stationarity in genetics is a generalization of the Mendelian laws. From the algebraic viewpoint it is very interesting to note that all idempotents of a Bernstein algebra are principal and hence primitive. In Section 3 below we obtain a necessary and sufficient condition, in terms of matrices, for a baric algebra to.be an associative algebra. In Section 4, we study the reticular properties of a baric algebra.

2. Preliminaries

DEFINITION. An algebra A is called baric if it admits a nontrivial homomorphism w over the underlying field (w is called a weight function of the algebra).

THEOREM. Let A be an n-dimensional algebra over K with weight homomorphism w. Then Ker w is an (n - 1)-dimensional ideal of A, and the factor algebra A/Ker w is isomorphic to the field K.

THEOREM. Let A be an n-dimensional algebra over R. The following conditions are equivalent:

(1) A is bark. (2) A has a basis {al,..., a,, } such that the multiplication constants defined by aiaj=EyS1yijta, satisfy C;=lyij,=l. (3)A has an (n - 1)-dimensional ideal N, and As g N. 600 R.

A is genetic if it admits a basis {c,, cr,. . . , c,, }, called the canonical basis, where the multiplication constants yijt satisfy yooo= 1, yojt = 0 for t < j, yijt = 0 for t < max(i, j). The constants Yooo,YOll? *. . ) Yonn are called train roots of the genetic algebra.

The genetic algebras are baric with weight function w(co) = 1, w(c,) = 0, t =l,...,n. A special train algebra (a baric algebra in which Ker w is nilpotent and the principal powers of Ker w are ideals of A) is genetic.

3. Associativity Let A be a (commutative) baric algebra over K of dimension n + 1, and let w be the weight homomorphism. Then A has an ideal Z of codimension one (the kernel of w), and there exists a basis { ao, a,,. . . , a, }, where {a r”“, a,} is a basis of Z and u. is an element of weight one. We will call such a basis a weight basis. So we have the following multiplication: aOaO = a, +Zr=, pja j, aOui = Cy,l pi .aj, and aiuk = X~=laikjaj, i # 0 # k. By commutativity oikj = lykij. We wili consider the matrices

A,=

aill . . . a. .f. (Yiln ‘15 Ai = cYihl . . . “ihj ” ’ (Yihn i=l,...,n. (Yin1 . .* ainj . .- ainn I>

Let c be the n x n matrix whose ith row is (PI,..., /?,)A,. We have:

THEOREM. The baric algebra A is associative if and only if for every i, j = 1 ,...> n

(i) A% = A, + C, (ii) A,A, = AiAo, (in) AiAj = A,A,. CONFERENCE REPORT 601

NOTE. In many cases (for instance in genetic algebras) we can choose the weight basis with the element a,, idempotent. Hence j3i = * * . = /3, = 0, and so C = 0. Therefore condition (i) reduces to At = A,.

4. Reticular Theory In any algebraic structure the knowledge of the lattice of substructures usually gives a great deal of information about the structure. For instance, Kaplansky [8] studies the nonassociative algebras with some additional properties in which every subspace is a subalgebra. Many other authors study other reticular conditions for nonassociative algebras. For baric algebras we can prove:

THEOREM 1. Let A be a baric algebra over K f Z/(2). Then every subspace of dimension one is a subalgebra if and only if the multiplication of any weight basis {a,,a,,...,a,,} is given by

a20 = a 0, aOai=+ai, aiaj=O V i,jE {l,..., n}. (9

THEOREM 2. Let A be a baric algebra over K # Z/(2). Then the following are equivalent:

(1) Every subspace of A is a subalgebra. (2) Every subspace of codimension one of A is a subalgebra. (3) Every subspace of dimmsim one is a subalgebra. (4) The multiplication of any weight basis of A is given by (!).

THEOREM 3. Every subalgebra of a baric algebra is again baric if and only if the multiplication of any weight basis is given by

aiaj C OLijkak, (!!) k=O where aiii = 1, 0 < i < n, and aijk = 0 for k -Cmax(i, j).

OPEN PROBLEMS.

(1) To continue studying the relations between genetic algebras and other classes of algebras. 602 R. BRU AND J. VITbRIA

(2) To obtain new reticular results for these algebras. For instance, to study the baric algebras with other conditions in their lattice of subalgebras or ideals. (3) To explore reticular theory and derivation algebras. Genetic interpretation.

With the above reticular results, Kaplansky in [8] determines the nonassociative algebras with a lot of symmetries, in the sense that the derivation algebra is large. The Lie algebra of derivations of a given algebra is an important tool for studying its structure, particularly in the nonassociative case. If A is an n-dimensional algebra with all products equal to zero, the dimension of the derivation algebra DerA must fall at least to n2 - n. Kaplansky determines the n-dimensional nonassociative algebras in which Der A is ( n2 - n)-dimensional. On the other hand, the derivations of a genetic algebra are studied, for instance, in [l], [2], [9], and [lo]. The genetic meaning of a derivation of a genetic algebra is given by Holgate in [7]. The task is then to obtain new results on the derivate algebra of a genetic algebra, using the reticular theory in the same fashion as in Kaplansky’s paper, and finally to interpret the obtained results genetically.

REFERENCES

1 R. Costa, On the derivations of gametic algebras for polyploidy with multiple alleles, Bol. Sot. Brad. Mat. 13:69-81 (1982). 2 R. Costa, On the derivation algebra of zygotic algebras for polyploidy of multiple alleles, Bol. Sot. Brud. Mat. 14:63-80 (1983). 3 I. Etherington, Genetic algebras, Proc. Roy. Sot. Edinburgh 59:242-258 (1939). 4 Non-associative algebra and the symbolism of genetics, Proc. Roy. Sot. Edinb&h 61:24-42 (1941). 5 P. Holgate, Jordan algebras arising in population genetics, Proc. Edinburgh Math. Sot. 15:291-294 (1967). 6 Genetic algebras satisfying Bernstein’s stationarity principle, J. London Math.’Sot. 9:613-623 (1975). 7 The interpretation of derivations in genetic algebras, Linear Algebra Appl.‘85:75-79 (1987). 8 I. Kaplansky, Algebras with many derivations, in Aspect of Mathematics and Its Applications, North-Holland, 1986, pp. 431-438. 9 Micah, Campos, Costa e Silva, Ferreira, and Costa, Derivations dam les algbbres gametiques, Cornm. Algebra 12:239-243 (1984). 10 Micah, Campos, Costa e Silva, and Ferreira, Derivations dans les algkbres gametiques II, Linear Algebra A&. 64:175-181 (1985). CONFERENCE REPORT 603

11 Wijn-Busekros, Bernstein Algebras, Arch. Math. 48, 1987. 12 R. D. Schafer, Structure of genetic algebras, Amer. J. Math. 71:121-135, 1949. 13 G. M. Piacentini, Gametic algebras of mutation and Jordan Algebras, Radic. Math. 13: 179- 186, 1980. 14 A. Micali and R. Duattara, Sur les algkbres de Jordan genetiques, Traveaux in Course, Ed. J. Dieudonn&, 1989.

GROUP AND MOORE-PENROSE INVERTIBILITY OF HANKEL AND TOEPLITZ MATRICES by M. C. GOUVEIA14

Zntroduction Inversion of Hankel and Toeplitz matrices arises in many applied problems in algebra, numerical and functional analysis, probability theory, system theory, etc. In this synopsis we present some methods of characterizing the group and Moore-Penrose inverses of Hankel and Toeplitz matrices. The results were obtained using the theory of such matrices together with the theory of generalized inverses. We refer to [6] and [8] for the notation and theory of Hankel and Toeplitz matrices, and to [2] and [lo] for the notation and theory of generalized inverses. Details and proofs will appear in the submitted paper [4].

1. Let %I,” be a Hankel matrix over a field F with characteristic (r, k). The theorem of Frobenius (see [8]) can be extended to nonsquare Hankel matrices, so that we can state that there exist matrices X,Y and Hankel matrices H, n, Hl,,, H,,, such that

is a full-rank factorization.

“Departamento de Matemitica, F.C.T.U.C., 3000 Coimbra,Portugal.